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Properties

Label 2-336-112.101-c2-0-24
Degree $2$
Conductor $336$
Sign $0.999 - 0.0152i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 − 1.47i)2-s + (1.67 − 0.448i)3-s + (−0.371 + 3.98i)4-s + (9.43 + 2.52i)5-s + (−2.91 − 1.86i)6-s + (−5.22 + 4.66i)7-s + (6.38 − 4.81i)8-s + (2.59 − 1.50i)9-s + (−8.96 − 17.3i)10-s + (1.67 − 0.449i)11-s + (1.16 + 6.82i)12-s + (2.65 + 2.65i)13-s + (13.9 + 1.43i)14-s + 16.9·15-s + (−15.7 − 2.95i)16-s + (10.0 + 5.83i)17-s + ⋯
L(s)  = 1  + (−0.673 − 0.739i)2-s + (0.557 − 0.149i)3-s + (−0.0929 + 0.995i)4-s + (1.88 + 0.505i)5-s + (−0.486 − 0.311i)6-s + (−0.745 + 0.666i)7-s + (0.798 − 0.601i)8-s + (0.288 − 0.166i)9-s + (−0.896 − 1.73i)10-s + (0.152 − 0.0408i)11-s + (0.0969 + 0.569i)12-s + (0.204 + 0.204i)13-s + (0.994 + 0.102i)14-s + 1.12·15-s + (−0.982 − 0.184i)16-s + (0.594 + 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0152i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.999 - 0.0152i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.999 - 0.0152i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.82708 + 0.0139070i\)
\(L(\frac12)\) \(\approx\) \(1.82708 + 0.0139070i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 1.47i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 + (5.22 - 4.66i)T \)
good5 \( 1 + (-9.43 - 2.52i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-1.67 + 0.449i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-2.65 - 2.65i)T + 169iT^{2} \)
17 \( 1 + (-10.0 - 5.83i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (2.63 - 9.85i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (23.3 - 13.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (12.8 - 12.8i)T - 841iT^{2} \)
31 \( 1 + (3.79 + 2.19i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.6 + 50.9i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 39.4T + 1.68e3T^{2} \)
43 \( 1 + (-25.0 - 25.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-66.4 + 38.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (89.7 - 24.0i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (23.1 + 86.3i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-12.4 + 46.4i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (11.3 + 42.2i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 73.7iT - 5.04e3T^{2} \)
73 \( 1 + (-16.3 + 28.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-21.6 - 37.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-2.53 - 2.53i)T + 6.88e3iT^{2} \)
89 \( 1 + (40.3 + 69.8i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 23.0iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.05393315249900117901291157960, −10.15378565748041168901446495362, −9.470444426988915856364513627695, −9.042252256213687737555827801544, −7.72796892415160166335474133992, −6.49427441011113078736453199040, −5.67621712577144252248575045492, −3.62439424599005434616270050568, −2.51865399226660883034403726662, −1.67606187718174003621489657034, 1.10368022481137698363478089756, 2.51925158339313203563172527231, 4.50805369685424559515978598676, 5.76848356331670421136479332781, 6.41642699192760091095368775044, 7.56026081573711616899766444109, 8.757621995181712368215566475927, 9.482235276833899297906462381923, 9.999802241021784730090337718385, 10.72153152216386075116115847840

Graph of the $Z$-function along the critical line